Half of the migrants - 830 - arrived on Chios and 698 others on Lesbos, two islands in the northeast Aegean which lie close to Turkey, according to the SOMP agency which is coordinating Athens' response to the migration crisis.
The continuing influx "creates a problem, and raises questions about the intent of all parties" in the agreement, SOMP spokesman Giorgos Kyritsis said.
The deal, reached at an EU-Turkey summit last Friday, aims at cutting off a route that enabled 850,000 people to pour into Europe last year, many of them fleeing conflict in Syria.
From midnight Saturday, all migrants landing on the Greek islands faced being sent back to Turkey.
For every Syrian sent back, the European Union will resettle one Syrian from the Turkish refugee camps where nearly 3mn people are living.
The idea is to reduce the incentive for Syrian refugees to board dangerous smugglers' boats to cross to Greece, encouraging them instead to stay in Turkish refugee camps to win a chance at resettlement in Europe.
The EU will also speed up talks on Ankara's bid to join the 28-nation bloc, double refugee aid to €6bn ($6.8bn), and give visa-free travel to Turks in Europe's Schengen passport-free zone by June.
All new arrivals are being taken to registration centres set up by the Greek authorities on five Aegean islands. Those seeking asylum will stay there while their application is considered by Greek and European officials, under the deal.
The EU has promised 2,300 officials and police to help the operation, according to the Greek government, but as of Monday none had arrived, the SOMP said.
"We hope that the upcoming Easter break will not be a factor in delaying their arrival, because the situation calls for an urgent response," Kyritsis said.
France and Germany have offered to send up to 600 police and asylum experts, while Romania said on Sunday that it would send 70.